Let $n\in\mathbb N$ and write $[n] = \{1,\dots,n\}$. Furthermore, for a (finite) set $X$ let $\binom{X}{k}$ denote the family of subsets of $X$ of size $k$. Now, define a $k$-cover of $[n]$ to be a family $K$ of subsets of $[n]$ so that $$\bigcup_{X\in K}\binom{X}{k} = \binom{[n]}{k}.$$ Furthermore let a $k,\ell$-cover of $[n]$ be a $k$-cover of sets of size $\ell$. Now let $\varphi(k,\ell,n)$ denote the minimal cardinality of any $k,\ell$-cover of $[n]$. Is there an exact expression for $\varphi$?
A few examples: $\varphi(1,2,n) = \lceil n/2\rceil$, since this is the number of edges in a matching with $n$ vertices, plus one if the number of vertices are uneven. Moreover, $\varphi(k,n,n)= 1$, obviously, as well as $\varphi(k,k,n)= \binom{n}{k}$. We have $\varphi(k,n-1,n) = k+1$, as every subset $X$ of size $k$ of $[n]$ is contained in exactly $n-k$ subsets of size $n-1$. Therefore any selection of $k+1$ subsets of size $n-1$ of $[n]$ contains $X$.
We also have $\varphi(2,3,5)\leq 6$, as $\{\{1,2,3\},\{2,3,4\},\{2,3,5\},\{1,3,4\},\{1,3,5\},\{2,4,5\}\}$ is a $2,3$-cover of $[5]$ of size $6$.
Moreover, $\varphi(k,n-2,n)\leq \binom{k+2}{2}$: let $K$ be a $k,n-1$-cover of $[n]$ of size $k+1$. To obtain a $k,n-2$-cover we can extract a $k,n-2$-cover for each $X\in K$. Note that for all $X,Y\in K$ with $X\neq Y$, $|X\cap Y| = n-2$, so for every set $X\in K$ there are $k$ subsets of size $n-2$ of $X$ contained in some other element of $K$, plus perhaps one which is contained in none of the others. Thus we obtain the upper bound $\binom{k+1}{2} + k+1 = \binom{k+2}{2}$.
And so forth. I imagine this concept appears in combinatorics, but I wouldn't know what to look for.